Uniqueness of topological solutions of self-dual Chern–Simons equation with collapsing vortices

We consider the following Chern–Simons equation,equation(0.1)Δu+1ε2eu(1−eu)=4π∑i=1Nδpiε,inΩ, where Ω is a 2-dimensional flat torus, ε>0ε>0 is a coupling parameter and δpδp stands for the Dirac measure concentrated at p. In this paper, we proved that the topological solutions of (0.1) are uniquely determined by the location of their vortices provided the coupling parameter ε   is small and the collapsing velocity of vortices piε is slow enough or fast enough comparing with ε. This extends the uniqueness results of Choe [5] and Tarantello [22]. Meanwhile, for any topological solution ψ   defined in R2R2 whose linearized operator is non-degenerate, we construct a sequence of topological solutions uεuε of (0.1) whose asymptotic limit is exactly ψ   after rescaling around 0. A consequence is that non-uniqueness of topological solutions in R2R2 implies non-uniqueness of topological solutions on torus with collapsing vortices.